Higher level quadrature amplitude modulation (QAM), together with coding, is an attractive constituent for next generation communication systems due to its high spectral efficiency [1]. Coding introduces redundancy to the information sent over the communication channels in order to correctly determine the data when errors occur in the transmission. Among coding techniques, block coding and convolutional coding are by far the most frequently used. In particular, in 1993, a new class of codes called Turbo codes, based on the parallel concatenation of two recursive convolutional codes separated by a turbo interleaver, was introduced by Berrou et al. [2], showing near Shannon limit performance. After the discovery of turbo codes, turbo-like systems based on block constituent codes (turbo product codes) [3] and on serial concatenation [4] were also introduced.
To cope with bursty noise, coding techniques are in general coupled with channel interleaving. At the transmitter the channel interleaver permutes the encoded bits before modulation and transmission. At the receiver the channel de-interleaver spreads the errors, thus making the decoder work more efficiently. As observed by Caire et al. [5] bit-interleaved coded modulation (BICM) achieves superior performance with respect to trellis coded modulation (TCM) [6] over fading channels.
In most applications, best results are obtained when soft-input decoders, e.g., soft-input convolutional decoders and turbo decoders, are employed at the receiver [5]. The decoder soft-input represents the information about the reliability of the coded bits and needs to be accurately estimated. The block which provides, from the received channel de-interleaved modulation symbols, a soft-information of the coded bits is called a soft-output (SO) de-mapper. The soft-input (SI) decoder can return a hard-output (SIHO decoder) or a soft-output (SISO decoder), depending on the considered system. In the latter case the SISO decoder is followed by a further SI block (e.g., iterative decoding [2], [3], [4]; turbo-equalization [7], [8]; iterative multi-user cancellation [9]; and iterative spatial interference cancellation [10], etc.).
Following the pioneering work of Zehavi on bit reliability [11], several authors have developed SO de-mapping algorithms specific for different communication systems [5], [12], [13].
In [14] Tosato and Bisaglia developed a simplified SO de-mapper for the 16-QAM and 64-QAM constellations of the HIPERLAN/2 standard [15]. Furthermore, they also suggested an extension to higher order QAM constellations. As noted by the authors, their proposed method (hereafter called a TOBI method) is also applicable to other systems/standards with minor changes due to, for example, different Gray coded patterns.
To better understand the addressed problem, the TOBI method is briefly introduced in the particular case of the HomePlug AV standard [16] that uses a turbo coding. The same considerations apply for other types of standards and other types of coding.
The HomePlug AV (HPAV) physical layer is shown in FIG. 1. At the transmitter, the information bits for HPAV data transmission, after scrambling, are turbo convolutionally encoded, bit-by-bit interleaved and then converted into symbols through a bit-mapper. The data symbols (belonging to unit power constellations) are serial-to-parallel converted for OFDM modulation carried out using N sub-carriers. Each of the N OFDM sub-carriers can be differently loaded, depending on the estimated signal-to-noise ratio (SNR) per sub-carrier, with one of the following modulations: BPSK, 4-QAM, 8-QAM, 16-QAM, 64-QAM, 256-QAM and 1024-QAM.
To reduce the complexity of the receiver, a suitable cyclic prefix is used to remove both inter-symbol and inter-channel interference (ISI and ICI). Finally, before an analog front end (AFE) block, which sends the resulting signal to the power-line channel, a peak limiter block is inserted to minimize the peak-to-average power ratio (PAPR).
At the receiver the signal after the AFE block is fed to an automatic gain control (AGC) and time synchronization block. For sake of simplicity, the block will be assumed to be ideal. After cyclic prefix removal and OFDM demodulation, and assuming that the cyclic prefix completely eliminates ISI and ICI, it is possible to insure a good or perfect synchronization. The channel is time invariant within each i-th OFDM symbol. The received signal yi[k] over the generic sub-carrier k can be written as:yi[k]=Gi[k]ai[k]+ni[k]  (1)wherein ai[k], Gi[k] and ni[k] are the transmitted symbol, the channel frequency response complex coefficient and the complex additive noise with variance σi2[k], over the generic k-th sub-carrier during the i-th OFDM symbol, respectively.
The output from the OFDM demodulator is then sent to a de-mapper, a de-interleaver, a turbo convolutional decoder and a de-scrambler to reconstruct and estimate the transmitted bits.
Let M=2m be the number of symbols {a=al+jaQ} of the generic constellation, so that m interleaved coded bits (of values 0 and 1) are mapped into the complex symbol. Let ai[k]=al,i[k]+jaQ,i[k] denote the symbol transmitted over the generic k-th sub-carrier during the i-th OFDM symbol, and {cil[k], . . . , cit[k], . . . , cim[k]} denote the corresponding coded bit sequence. We note here that in the HomePlug AV, for square QAM constellations, the sub-sequence {ci1[k], ci2[k], . . . , ci−2[k]} is mapped into the real part al,i[k] of the QAM symbol transmitted over the generic k-th sub-carrier while the sub-sequence {cim/2+1[k], cim/2+2[k], . . . , cim[k]} is mapped into the corresponding imaginary part. For each bit ci1[k] the constellation is split into two partitions of complex symbols associated to the coded bits sequences with a ‘0’ in position l, namely Sl(0)and the complementary partition Sl(1).
The log-likelihood ratio (LLR) of the decision bit ci1{circumflex over ([)}k] from the de-mapper is [14]
                                                                                          λ                                      i                    l                                                  ⁡                                  [                  k                  ]                                            =                              ln                ⁡                                  (                                                            P                      ⁡                                              [                                                                                                            c                                                              i                                l                                                                                      ⁡                                                          [                              k                              ]                                                                                =                                                      1                            |                                                                                          y                                i                                                            ⁡                                                              [                                k                                ]                                                                                                                                    ]                                                                                    P                      ⁡                                              [                                                                                                            c                                                              i                                l                                                                                      ⁡                                                          [                              k                              ]                                                                                =                                                      0                            |                                                                                          y                                i                                                            ⁡                                                              [                                k                                ]                                                                                                                                    ]                                                                              )                                                                                                        =                              ln                (                                                                            ∑                                              α                        ∈                                                  S                          l                                                      (                            1                            )                                                                                                                ⁢                                          exp                      (                                              -                                                                                                                                                                                                                                                y                                    i                                                                    ⁡                                                                      [                                    k                                    ]                                                                                                  -                                                                  α                                  ⁢                                                                                                                                          ⁢                                                                                                            G                                      i                                                                        ⁡                                                                          [                                      k                                      ]                                                                                                                                                                                                                          2                                                                                2                            ⁢                                                          σ                              2                                                                                                                          )                                                                                                  ∑                                              α                        ∈                                                  S                          l                                                      (                            0                            )                                                                                                                ⁢                                          exp                      (                                              -                                                                                                                                                                                                                                                y                                    i                                                                    ⁡                                                                      [                                    k                                    ]                                                                                                  -                                                                  α                                  ⁢                                                                                                                                          ⁢                                                                                                            G                                      i                                                                        ⁡                                                                          [                                      k                                      ]                                                                                                                                                                                                                          2                                                                                2                            ⁢                                                          σ                              2                                                                                                                          )                                                                      )                                                                        (        2        )            wherein Sl(x) is the set of symbols for which the l-th bit is x (x=0, 1).Approximating the above formula with the Max-Log approximation
                                                        ∑              j                        ⁢                          exp              ⁡                              [                                  x                  j                                ]                                              ≈                      exp            (                                          max                j                            ⁢                              x                j                                      )                          ⁢                                  ⁢                  equation          ⁢                                          ⁢                      (            2            )                    ⁢                                          ⁢          can          ⁢                                          ⁢          be          ⁢                                          ⁢          written          ⁢                                          ⁢          as                                    (        3        )                                                                    λ                              l                i                                      ⁡                          [              k              ]                                ≈                                    (                                                                    min                                          α                      ∈                                              S                        l                                                  (                          0                          )                                                                                                      ⁢                                      (                                                                                            w                          i                                                ⁡                                                  [                          k                          ]                                                                    2                                        )                                                  -                                                      min                                          α                      ∈                                              S                        l                                                  (                          1                          )                                                                                                      ⁢                                      (                                                                                            w                          i                                                ⁡                                                  [                          k                          ]                                                                    2                                        )                                                              )                        ⁢                                                                                                                        G                      i                                        ⁡                                          [                      k                      ]                                                                                        2                                            2                ⁢                                  σ                  2                                                                    ⁢                                  ⁢        where                            (        4        )                                                      w            i                    ⁡                      [            k            ]                          =                                                                                                              y                    i                                    ⁡                                      [                    k                    ]                                                                                        G                    i                                    ⁡                                      [                    k                    ]                                                              -              α                                            =                                                                                  z                  i                                ⁡                                  [                  k                  ]                                            -              α                                                                      (        5        )            and zi[k]=yi[k]/Gi[k] represents the one-tap equalized received signal, over a generic sub-carrier k. Let us introduce the notation
                                          D                          i              l                                ⁡                      [            k            ]                          =                              1            4                    ⁢                      (                                                            min                                      α                    ∈                                          S                      l                                              (                        0                        )                                                                                            ⁢                                  (                                                                                    w                        i                                            ⁡                                              [                        k                        ]                                                              2                                    )                                            -                                                min                                      α                    ∈                                          S                      l                                              (                        1                        )                                                                                            ⁢                                  (                                                                                    w                        i                                            ⁡                                              [                        k                        ]                                                              2                                    )                                                      )                                              (        6        )            then equation (4) becomes
                                          λ                          i              l                                ⁡                      [            k            ]                          ≈                                            D                              i                l                                      ⁡                          [              k              ]                                ⁢                                    2              ⁢                                                                                                            G                      i                                        ⁡                                          [                      k                      ]                                                                                        2                                                                    σ                i                2                            ⁡                              [                k                ]                                                                        (        7        )            
The values given by equation (7) are input to the decoder, a sample architecture of which is depicted in FIG. 2, that processes them and decides which bits have been transmitted.
Before the TOBI method, the realization of the soft-output (SO) de-mapper for BICM systems was typically handled by methods which try to exactly compute equation (2). This involved expressions in which quotients of sum of exponential functions were computed although, at the end, some approximations were given [13]. Let us define:wl,i[k]={zi[k]}−al=zl,i[k]−al,wQ,i[k]={zi[k]}−aQ=zQ,i[k]−aQ  (8)where the notations {•} and {•} designate the real and the imaginary parts of their argument, respectively. In [14] it is demonstrated that, for square QAM constellations the computation of equation (7) can be reduced to
                                                                        λ                                  i                  l                                            ⁡                              [                k                ]                                      ≈                                                            D                                      I                    ,                                          i                      l                                                                      ⁡                                  [                  k                  ]                                            ⁢                                                2                  ⁢                                                                                                                                    G                          i                                                ⁡                                                  [                          k                          ]                                                                                                            2                                                                                        σ                    i                    2                                    ⁡                                      [                    k                    ]                                                              ⁢                                                          ⁢              l                                =                      1            ,            2                          ,        …        ⁢                                  ,                  m          /          2                                    (        9        )                                                                                                      λ                                      i                    l                                                  ⁡                                  [                  k                  ]                                            ≈                                                                    D                                          Q                      ,                                              i                        l                                                                              ⁡                                      [                    k                    ]                                                  ⁢                                                      2                    ⁢                                                                                                                                                G                            i                                                    ⁡                                                      [                            k                            ]                                                                                                                      2                                                                                                  σ                      i                      2                                        ⁡                                          [                      k                      ]                                                                      ⁢                                                                  ⁢                l                                      =                                          m                /                2                            +              1                                ,                                    m              /              2                        +            2                    ,          …          ⁢                                          ,          m                ⁢                                  ⁢        where                            (        10        )                                                                    D                              I                ,                                  i                  l                                                      ⁡                          [              k              ]                                =                                    1              4                        ⁢                          (                                                                    min                                                                  α                        I                                            ∈                                              S                                                  I                          ,                          l                                                                          (                          0                          )                                                                                                      ⁢                                      (                                                                                            w                                                      I                            ,                            i                                                                          ⁡                                                  [                          k                          ]                                                                    2                                        )                                                  -                                                      min                                                                  α                        I                                            ∈                                              S                                                  I                          ,                          l                                                                          (                          1                          )                                                                                                      ⁢                                      (                                                                                            w                                                      l                            ,                            i                                                                          ⁡                                                  [                          k                          ]                                                                    2                                        )                                                              )                                      ⁢                                  ⁢                              l            =                          1              ,              2                                ,          …          ⁢                                          ,                      m            /            2                                              (        11        )                                                                    D                              Q                ,                                  i                  l                                                      ⁡                          [              k              ]                                =                                    1              4                        ⁢                          (                                                                    min                                                                  α                        Q                                            ∈                                              S                                                  Q                          ,                          l                                                                          (                          0                          )                                                                                                      ⁢                                      (                                                                                            w                                                      Q                            ,                            i                                                                          ⁡                                                  [                          k                          ]                                                                    2                                        )                                                  -                                                      min                                                                  α                        Q                                            ∈                                              S                                                  Q                          ,                          l                                                                          (                          1                          )                                                                                                      ⁢                                      (                                                                                            w                                                      Q                            ,                            i                                                                          ⁡                                                  [                          k                          ]                                                                    2                                        )                                                              )                                      ⁢                                  ⁢                              l            =                                          m                /                2                            +              1                                ,                                    m              /              2                        +            2                    ,          …          ⁢                                          ,          m                                    (        12        )            S1,l(x) contains the real parts of the complex symbols of subset Sl(x) for x=0, 1 and l=1, 2, . . . , m/2 and SQ,l(x) contains the imaginary parts of the complex symbols of subset Sl(x) for x=0, 1 and l=m/2+1, m/2+2, . . . , m.
As explained in [14] the main simplification of (11) and (12) with respect to (6), lies in the fact that the two dimensional Euclidean distances from M constellation points of (6) reduce to one-dimensional Euclidean distances from √{square root over (M)} points of (11) and (12) allowing a significant decrease in the computational complexity.
Hereafter, the method to estimate the LLRs based on (9), (10), (11) and (12) will be referred to the Max-Log method. The Max-Log method, although it introduces significant simplifications with respect to the computation of (2), is cumbersome especially for higher order QAM constellations. For this reason, in [14], further simplified expressions are given. Below, the TOBI expressions are derived for the higher HomePlug AV constellations, namely: 64-QAM, 256-QAM and 1024-QAM, taking into account the HomePlug AV Gray pattern and the normalization factors.